Optimal Design of Energy-Efficient Millimeter Wave Hybrid Transceivers for Wireless Backhaul
OOptimal Design of Energy-Efficient MillimeterWave Hybrid Transceivers for Wireless Backhaul
Andrea Pizzo and Luca Sanguinetti
Dipartimento di Ingegneria dell’Informazione, University of Pisa, Pisa, Italy
Abstract —This work analyzes a mmWave single-cell network,which comprises a macro base station (BS) and an overlaid tierof small-cell BSs using a wireless backhaul for data traffic. Welook for the optimal number of antennas at both BS and small-cell BSs that maximize the energy efficiency (EE) of the systemwhen a hybrid transceiver architecture is employed. Closed-formexpressions for the EE-optimal values of the number of antennasare derived that provide valuable insights into the interplaybetween the optimization variables and hardware characteristics.Numerical and analytical results show that the maximal EE isachieved by a ’close-to’ fully-digital system wherein the numberof BS antennas is approximately equal to the number of servedsmall cells.
I. I
NTRODUCTION
Millimeter Wave (mmWave) communications suffer fromhigh atmospheric absorption, rain and foliage attenuation,penetration and reflection losses, which essentially restricttheir use to line-of-sight (LoS) indoor-to-indoor or outdoor-to-outdoor communications over relatively short distances [1].Nevertheless, recent theoretical considerations and measure-ment campaigns have provided evidence that outdoor cellswith up to m cell radii are viable if transmitters andreceivers are equipped with sufficiently large antenna ar-rays along with beamforming [2], [3]. However, large arraysbeamforming poses several implementation challenges mostlybecause of hardware limitations that make hard to have adedicated baseband and radio frequency (RF) chain for eachantenna. Analog solutions arise in early works for mmWavesystems for their ease of implementation and power saving [4],[5] at the price of single-stream transmissions that substantiallylimit the system spectral efficiency. To combine the benefits ofanalog and digital architectures, hybrid beamforming schemeshave gained a lot of interest [6].A hybrid beamformer is made up of a low-dimensionalbaseband precoder followed by a high-dimensional RF beam-former. The latter is fully implemented by low-cost and powerefficient analog phase shifters. Interestingly, in [7] the authorsprovide necessary and sufficient conditions to realize any fully-digital beamformer by using a hybrid one. The literature onhybrid beamforming schemes is relatively vast. In [8] and[9], a point-to-point multiple-input multiple-output (MIMO)system is considered while the downlink of a multi-user settingis investigated in [10] using single-antenna receivers and asingle-stream transmitter with a RF chain per user. In [11], theauthors consider the more realistic case of imperfect channelstate information due to the limited feedback of the returnchannel. All the aforementioned works are mainly focusedon increasing the system spectral efficiency. There exist alsosome literature looking at reducing the power consumption. Examples towards this direction can be found in [12] and[13]. In particular, [12] proposes the use of low-cost switchesfor implementing antenna selection schemes whereas [13]provides algorithms for selecting a subset of antennas. In[14], different hybrid architectures are compared in terms ofboth spectral and energy efficiency (EE), defined as the ratiobetween throughput and power consumption. Switching-basedsolutions are found to performs poorly compared to both fully-digital and hybrid schemes.In addition to mobile communications, the main use casesof mmWave communications are wireless local area networks(WLANs) based on the IEEE 802.11ad standard as well aswireless backhaul in the unlicensed 60 Ghz band as a cost-efficient alternative to wired solutions. Wireless backhaul atmmWave bands is considered in [5], wherein the design ofbeam alignment techniques is investigated for a single-cellpoint-to-point network using an analog-only transceiver. Alongthis line of research, this work focuses on the downlink of asingle-cell network in which a given number of multiple small-cell BSs exchange data with a macro BS through wirelessbackhaul, using a low-cost hybrid transceiver architecture [10],[11]. Our goal is to find respectively the optimal number N and M of antennas at the BS and each small-cell BS in order tomaximize the EE. To this end, we first model the consumedpower of a hybrid transceiver architecture at mmWave andthen derive closed-form EE-optimal values for M and N .These expressions provide valuable design insights into theinterplay between system parameters and different componentsof the consumed power model. This work is inspired to theframework developed in [15], which however deals with theEE of massive MIMO networks and thus does not fit networksoperating at mmWave frequencies.The remainder of this work is organized as follows. Nextsection introduces the system model under a LoS channelpropagation model and formulates the EE maximization prob-lem. Section III develops the power consumption model of thehybrid transceiver network as a function of different systemparameters. The EE-optimal number of antennas are computedin Section IV. Numerical results are given in Section V tovalidate the theoretical analysis. The numerical results are thenextended to a more realistic clustered mmWave channel modelin Section VI. Conclusions are drawn in Section VII.II. N ETWORK MODEL AND P ROBLEM S TATEMENT
A. Network model
We consider a two-tier network, which comprises a macroBS equipped with N antennas and an overlaid tier of K small-cell BSs (selected from a larger set) endowed with M antennas a r X i v : . [ c s . I T ] M a r Bpre-coder HPAHPARF chain LNALNA f RF1 f RF K F BB DACDACDACDAC
NN NKs K s RF chain ADCADC
M K w H1 w H K LNALNA RF chain M RF chain H H K LPFLPFLPFLPF LPFLPF ADCADCLPFLPF
Fig. 1: Transceiver chain architecture.and using a mmWave wireless backhaul link over a bandwidth B . We assume that the small-cell BSs are deployed so as tobe in visibility with the macro BS. Due to the high absorptionof scattered rays and the use of large antenna arrays (thatcreate very narrow beam) at mmWave bands, a LoS modelcan be reasonably adopted for the propagation channel of eachtransmission link. In these circumstances, the channel matrix H k ∈ C N × M between the BS and small-cell BS k can bemodeled as: H k = √ α k a N ( φ k ) a H M ( θ k ) (1)where a N ∈ C N × and a M ∈ C M × account, respectively,for the array manifolds of the BS and small-cell BSs with φ k and θ k being the angles of departure and arrival of the LoSlink k . The parameter α k describes the macroscopic pathlossand is computed as α k = 10 − l k, dB / with [5] l k, dB = 32 . f c + 10 log ( d k ) β + Ad k + ξ (2)where f c [GHz] is the carrier frequency, β is the pathlossexponent, d k [km] denotes the distance between the BS andsmall-cell BS k , A accounts for the oxygen absorption andrainfall effect whereas ξ ∼ CN (0 , σ ξ ) is the shadowing beingcomplex circularly symmetric Gaussian with variance σ ξ .Channel acquisition at mmWave bands is generally a chal-lenging task due to the large number of antennas and thehigh bandwidth. However, if an uniform linear array (ULA) isadopted at both sides, the channel acquisition problem simplyreduces to estimating the sets of directions { θ k , φ k } andpathlosses { α k } cutting down the number of unknowns from ( N M ) K to K . If mmWave communications are used forwireless backhaul, then channel estimation simplifies furtherdue to the absence of mobility and the favorable deploymentof the macro BS and small-cell BSs. In these circumstances,perfect channel state information seems to be a reasonableassumption (e.g., [5] and [17]). Based on this observation, inthis work we assume perfect knowledge of { θ k , φ k , α k } . Tolimit the implementation costs [11], we assume that a two-stage linear hybrid precoding scheme is employed at the BS Observe that the LoS condition is also valid in highly dense mmWavenetworks, where having links in visibility is more likely to happen [16]. and that a RF linear combiner is used at each small-cell BS(see Fig. 1). In particular, the BS employs a baseband precoder F BB = [ f BB1 , · · · , f BB K ] ∈ C R × K followed by a RF precoder F RF = [ f RF1 , · · · , f RF R ] ∈ C N × R with K ≤ R ≤ N being thenumber of RF chains. The transmitted vector x ∈ C N is thusgiven by x = F RF F BB s where s ∈ C K is the data vectorsuch that E { ss H } = P/N I K with P being the transmittedpower. Hereafter, we assume that R = K , i.e. one stream persmall-cell BS is allocated.At small-cell BS k , the received signal is linearly processedthrough the RF combiner w k to obtain: y k = w H k H H k x + w H k n k (3)where n k ∼ CN ( , σ I M ) is the thermal noise with σ = BN N F [W] while N [W/Hz] and N F being the noisepower spectral density and noise figure, respectively. TheRF combiners { w k } and precoders { f RF k } are implementedusing analog phase shifters. Under the assumption of perfectknowledge of { θ k , φ k } , we have that w k = a M ( θ k ) and f RF k = a N ( φ k ) . Therefore, y k reduces to: y k = ( M N )¯ h H k F BB s + a H M ( θ k ) n k (4)where ¯ h H k = √ α k N a H N ( φ k ) F RF is the effective channel seenfrom small-cell BS k after receive combining. The BB pre-coder F BB is designed according to a zero-forcing (ZF)criterion so as to completely remove the interference amongsmall-cell BSs [11]. This leads to F BB = ( ¯ H H ) − where ¯ H H = [¯ h , . . . , ¯ h K ] H = N D / ( F H RF F RF ) with D =diag ( α , . . . , α K ) . Plugging F BB = ( ¯ H H ) − into (4) yields y k = ( M N ) s k + a H M ( θ k ) n k . (5)Note that the inverse of ¯ H H exists as long as φ l − φ k = 0 for k, l = 1 , . . . , K , which always occurs in practice if the servedsmall-cell BSs are properly selected. B. Problem statement
The aim of this work is to compute the values of (
N, M )that, for a given number K of small-cell BSs, maximize theE of the network given by: EE =
ThroughputConsumed Power [ bit/Joule ] (6)which stands for the number of bits that can be reliablytransmitted per unit of energy. From (5), the throughput ofthe considered network is easily found as:Throughput = BK log (1 + M N γ ) [ bit/s ] (7)with γ = P/σ . Observe that we have neglected the pre-logfactor that should take into account the signaling overhead forchannel estimation, due to the stationarity of the investigatednetwork [15]. The consumed power is computed as [15]Consumed Power = η − P x + P CP [ W ] (8)where P x is the transmit power, η ≤ is the power amplifier(PA) efficiency and P CP accounts for the power consumed bythe circuitry.III. P OWER CONSUMPTION MODEL
A reasonable circuit power consumption model for a genericBS in a cellular network is as follows [15] P CP = P FIX + P TC + P LP + P CE + P C / BH (9)where P FIX accounts for the fixed power consumption of thesystem, P TC of the transceiver chain (at both BS and small-cell sides), P CE of the channel estimation process, P LP of thelinear processing, P C / BH of the coding at BS and of the load-dependent backhauling cost. Next all the above terms will beexplicated as a function of all the system parameters in Table Itaken for a reference carrier frequency of f c = 60 Ghz.
A. Transmitted power
The average transmit power is given by P x = E {k x k } where the expectation is taken with respect to the set ofdistances d = [ d , · · · , d K ] and AoDs φ = [ φ , · · · , φ K ] ,and thus, can be computed as P x = tr (cid:16) E n ss H o E n F H BB F H RF F RF F BB o(cid:17) = PN tr (cid:16) E n ( ¯ H H ) − ( N D − / ¯ H H )( ¯ H H ) − o(cid:17) = N P tr (cid:16) E n D − ( F H RF F RF ) − o(cid:17) = N P K X k =1 E d { α − k } E φ (cid:26)h ( F H RF F RF ) − i k,k (cid:27) ( a ) = N P K ¯ α E φ n(cid:2) P − (cid:3) k,k o (10)where ( a ) follows from assuming that any small-cell loca-tion is drawn from the same spatial distribution such that ¯ α = E d { α − k } . Also, we have defined for notational simplicity P = F H RF F RF ∈ C K × K . A possible way to deal withthe computation of E φ (cid:8)(cid:2) P − (cid:3) k,k (cid:9) is to make use of the Kantorovic inequality [11], which reads (exploiting the factthat [ P ] k,k = 1 ) (cid:2) P − (cid:3) k,k ≤ P ] k,k (cid:0) κ ( P ) + κ ( P ) − + 2 (cid:1) = 14 N (cid:0) κ ( P ) + κ ( P ) − + 2 (cid:1) (11)where κ ( P ) = κ ( F RF ) and κ ( F RF ) = k F RF kk F † RF k stands for the 2-norm condition number of the Vandermondematrix with entries [ F RF ] n,k = z nk for n = 0 , · · · , N − and nodes { z k } Kk =1 = e jπ sin( φ k ) for normalized antennaspacing ∆ / (2 πf c ) = 1 / . Computing κ ( F RF ) is a challenge,especially because the analysis must be valid for the entirerange of antennas. Vandermonde matrices with positive realnodes z k ∈ R + are well-known to be ill-conditioned [23] - thecondition number grows at least exponentially with the numberof nodes K . However, if the nodes are complex-valued, it ispossible to lower this growth to polynomial [24] and evenachieve perfect conditioning choosing the nodes to be roots ofunity [25]. In [26], the authors generalize this result to nodesthat are close enough to the unit circle (not necessarily on theunit circle) and not so close to each other, while having N large enough. In particular, it turns out that if | z k | = 1 and N > K − δ then [26] ≤ κ ( F RF ) ≤ δ K − N − δ K − N (12)with δ = min j = k | z j − z k | accounting for the worst-case nodeseparation. Thus, in order for the Vandermonde matrix F RF to be nearly perfect conditioned we better impose N ≫ K − δ . (13)To get some insight into how much large N should be, weconsider a uniformly spaced small-cell deployment on the rightside quadrants and evaluate δ . If the small-cell BSs are suchthat { φ k } Kk =1 = − πK ⌊ K ⌋ + πK then δ = | z ⌊ K ⌋ − z ⌊ K ⌋− | = | − e jπ sin( φ ) | = 2 (cid:12)(cid:12)(cid:12) sin (cid:16) π (cid:16) πK (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) (14)from which it follows that, when K grows large, δ can be well-approximated with π /K (using first order Taylor expansion).Plugging π /K into (13) leads to N ≫ K ( K − /π .This means that, for K sufficiently large, the value of N forachieving good conditioning for F RF is given by N ≥ λπ K = µ K (15)with λ ≥ being a design parameter. Under this condition,by using (11) and (15) into (10) we have that P x can bereasonably approximated as P x = P K ¯ α for N ≥ µ K . (16)Fig. 2 illustrates κ ( F RF ) as a function of N for different valuesof K and uniformly spaced nodes, i.e. { φ k } Kk =1 = − πK ⌊ K ⌋ + ABLE I: Network and system parameters at 60GHz.
Parameter Description Value Parameter Description Value P LNA
Power consumed by low noise amplifier [18] 39 [mW] L BH Power used by backhauling per bit/s [15] 250 [mW/Gbit/s] P HPA
Power consumed by the high-power-amplifier [18], [19] 138 [mW] T c Coherence time [20] 10 [s] P DC Power absorbed by the down conversion stage [21] 47.3 [mW] ∆ Normalised antenna separation 0.5 P UC Power absorbed by the up conversion stage [19] 49 [mW] σ ξ Shadowing variance [5] 8 [dB] P ADC
Power needed to run the analog-to-digital converter [12] 200 [mW] A Oxigen and rainfall absorption [5] 25 [dB] P DAC
Power needed to run the digital-to-analog converter [22] 110 [mW] κ Path-loss exponent [5] 2.2 P C Power consumed by the combiner [18] 19.5 [mW] N Noise power spectral density [5] -174 [dBm] P PS Power required to commute phase shifter [18] 30 [mW] d Distance BS to small-cell BSs [5] 150 [m] L BS Computational efficiency at the BS [15] 20 [Gflops/W] N F Noise figure [5] 6 [dB] L SC Computational efficiency at the small-cell BSs [15] 5 [Gflops/W] B Transmission bandwidth [5] 2 [Ghz] L C Power consumed performing coding per bit/s [15] 100 [mW/Gbit/s] f c Carrier frequency [5] 60 [Ghz] L D Power consumed performing decoding per bit/s [15] 800 [mW/Gbit/s] η High power amplifier efficiency [15] 0.375
10 15 20 25 302468101214161820 κ ( F R F ) Number of antennas N K = 2 K = 6 K = 8 Fig. 2: Condition number of F RF ( φ ) versus N for K = 2 , and . πK . As seen, κ ( F RF ) tends to unity when N grows for any K .Also, it can be seen numerically that λ = 1 is already enoughto satisfy condition (15) when the nodes (small-cell BSs) areproperly selected. A similar behavior is observed for AoDsuniformly distributed φ k ∈ U [ − π/ , π/ [27]. B. Transceiver Chain
The transceiver architecture of the investigated network issketched in Fig.1. We assume that both the BS and the setof small-cell BSs make use of (at least) 5-bit passive phaseshifters (PSs) that emulate the arbitrary angles matching atRF [12]. Each small-cell BS consists of a single RF chainconnected through a combiner to M parallel front-end (FE)receivers, one for each receive antenna. Each FE receiver iscomposed of a low-noise amplifier (LNA) followed by a phaseshifter, while an RF chain hosts a couple (I/Q) of analog-to-digital converters (ADCs), and a down conversion stagethat includes a mixer, a voltage controlled oscillator and abaseband buffer [21]. Therefore, the power consumption ofthe transceiver chain at each small-cell BS can be computedas P SCTC = M ( P LNA + P PS ) | {z } Front-end + P DC + P ADC + P C | {z } RF chain (17)where P LNA accounts for the power consumption of eachLNA, P PS of each PS, P DC of the down-conversion, P ADC of the ADC and P C of the combiner. On the other hand, the BS transceiver consists of K RFchains each one fetching a rake of N PSs that drive the phasesof N antennas, each one with a high power amplifier (HPA).Each RF chain has a pair (I/Q) of digital-to-analog converters(DACs) plus a combiner as well as an up-conversion stageincluding filtering and amplifying. Therefore, we have that P BSTC = N ( KP PS + P HPA + P C ) | {z } Front-end + K ( P UC + P DAC ) | {z } RF chain (18)Therefore, the total amount of consumed power in thetransceiver chain is P TC = P BSTC + KP SCTC = p RF + p SCFE M + p BSFE N (19)where p RF = K ( P DC + P ADC + P C + P UC + P DAC ) accountsfor the power consumption of the RF chain at both sides,whereas p SCFE = K ( P LNA + P PS ) and p BSFE = KP PS + P HPA + P C of the FEs at the small-cells BS and BS, respectively. C. Linear Processing
The power consumed by linear processing accounts for allthe operations performed in the digital domain at the macroBS. This be quantified as P LP = P LP − T | {z } Transmission + P LP − P | {z } Precoder computation (20)where P LP − T accounts for the total power consumed bydownlink transmission of payload samples whereas P LP − P is the power required for the computation of F BB . Due tothe stationarity of the investigated network, the latter can beneglected since it is computed once for all. This amounts tosaying that P LP − P = 0 . The computation of F BB s requires atotal of K (2 K − complex operations per sample. Denotingby L BS the computational efficiency of the BS [flops/W], wehave that P LP = B K (2 K − L BS . (21) D. Coding/Decoding and Backhauling
Load-dependent power costs are given by coding/decodingand backhauling. In the downlink, the BS applies chan-nel coding and modulation to K sequences of informationsymbols and each small-cell BS applies some suboptimalfixed-complexity algorithm for decoding its own sequence.The opposite is done in the uplink. The power consumptionccounting for these processes is proportional to the numberof bits. The backhaul is used to transfer uplink/downlink databetween the BS and the core network. The power consumptionof the backhaul is commonly modeled as the sum of two parts:one load-independent (included in the fix power consump-tion) and one load-dependent (proportional to the throughput).Therefore, the power consumption for coding/decoding andbackhauling processes can be computed as P C / BH = L B BK log (1 + M N γ ) (22)where L B = L C / D + L BH with L C / D and L BH beingthe operational costs for coding/decoding and backhauling,respectively. IV. EE OPTIMIZATION
Plugging (7)-(9) and (16)-(22) into (6), the EE optimizationproblem can thus be formulated as arg max ( M,N ) ∈ Z ++ EE(
M, N, K ) s.t. N ≥ µ K (23)with EE = BK log (1 + γM N )¯ P FIX + p SCFE M + p BSFE N (24)and ¯ P FIX = p RF + P FIX + P x η − + P LP . (25)In the following, we aim at solving (23) for fixed systemparameters as given in Table I. In doing so, we first derivea closed-form expression for the EE-optimal value of both M and N when the other one is fixed. This does not only bringindispensable insights into the interplay between ( M, N ) andthe system parameters, but provides the means to solve theproblem by a sequential optimization algorithm. A. Optimum number of small-cell BS antennas
We begin by deriving the optimal number of small-cell BSantennas M while N is fixed. Applying [15, Lemma 3], itreadily follows that: Lemma 1.
Assume N is given, then the optimal M can becomputed as M ⋆ = ⌊ x ⋆ ⌉ with x ⋆ = e W (cid:0) γe c c − e (cid:1) +1 − γN (26) and c = N (cid:16) ¯ P FIX + p BSFE N (cid:17) , c = p SCFE and ⌊·⌉ as the nearestinteger projector.
The above result provides explicit guidelines on how toselect M in a hybrid mmWave system for maximal EE. Noticethat the term c depends, through ¯ P FIX , on p RF , which ac-counts for the RF chain power consumption of the transceiverarchitecture, and also on the front-end power consumption p BSFE at the BS. Using the typical values of Table I, it turns outthat c is on the order of hundreds of Watt for a relativelysmall number of antennas N . Larger values are obtained if N increases. On the other hand, c does not depend on N and takes values in the range of Watt, since it depends only onthe power consumed by the small-cell BSs for the front-end.Therefore, we can reasonably assume that, for typical valuesof system parameters, c /c ≫ such that e W ( r )+1 can beapproximated with r and x ⋆ reduces to x ⋆ ≈ N e c c = ¯ P FIX + p BSFE
Ne p
SCFE . (27)Using the above result and the power consumption expressionsprovided in Section III, the following corollary is found: Corollary 1. If N and K grow large, then M ⋆ increasesmonotonically as: M ⋆ ≈ (cid:22) ξ + p BSFE p SCFE N (cid:25) (28) with ξ = (cid:0) p RF + P FIX + BL BS K (cid:1) /p SCFE , p SCFE and p BSFE as in (17) and (18) , respectively.
From the above corollary, it follows that M ⋆ is monotoni-cally increasing with P FIX as well as with K and N . Usingthe values reported in Table I, it turns out that p BSFE /p SCFE < ,meaning that M ⋆ grows at a slower pace than N . Also, theterm ξ indicates that M ⋆ increases linearly with p RF , i.e., thepower consumed by the FE at both the BS and small-cell BSs. B. Optimum number of BS antennas
We now look for the value of N that maximizes the EE in(23). Still, by using [15, Lemma 3] and exploiting the pseudoconcavity of the objective function, the following result isobtained: Lemma 2.
Assume M is given, then the optimal N is givenby N ⋆ = ⌊ z ⋆ ⌉ with z ⋆ = max e W (cid:16) γe d d − e (cid:17) +1 − γM , µ K (29) and d = M (cid:16) ¯ P FIX + p SCFE M (cid:17) , d = p BSFE , µ K as in (15) and ⌊·⌉ as the nearest integer projector. As for M , we have that z ⋆ can be reasonably approximatedas z ⋆ ≈ N e d d from which it follows that: Corollary 2. If M and K grow large, then N ⋆ increasesmonotonically as: N ⋆ ≈ (cid:22) max (cid:26) ξ + p SCFE p BSFE
M, µ K (cid:27)(cid:25) (30)In agreement with the results of Corollary 1, we have that N ⋆ grows at faster pace than M since p SCFE /p BSFE > as itfollows using the values of Table I. Therefore, using largerarrays at the BS rather than at small-cell BSs seems to be amore natural choice for maximal EE. The interested reader is referred to [28] for further details on the inequal-ities and approximations involving the Lambert function. . Sequential Optimization of
M, N
Using Lemmas 1 and 2, a sequential optimization algorithmto solve (23) operates as follows:1) Optimize M for a fixed N using Lemma 1;2) Optimize N for a fixed M using Lemma 2;3) Repeat 1)–2) until convergence is achieved.This algorithm converges since the EE is a non-decreasingmonotone function of ( M, N ) and bounded above. The mono-tonicity is ensured by the pseudo concavity of (24). Indeed, thenumerator is non-negative, differentiable, and concave, whilethe denominator is differentiable and affine, and so convex.V. N UMERICAL RESULTS
Numerical results are now used to validate the analysis. Weconsider a single-cell scenario as described in Section II witha macro BS, operating at f c = 60 GHz over a bandwidthof B = 2 GHz placed at the center of the cell and servingsimultaneously K small-cell BSs, with a distance d = 150 m from the BS. To avoid ambiguity in the spatial domain,the small cells are angularly displaced on the right half-space centered on the BS. The channel parameters and allof the terms introduced in Section III are listed in Table I. Tomake the numerical results as realistic as possible, the samefabrication technology (65nm CMOS ) is used for the circuitparameters (e.g. [18] and [12]), while the linear processing andthe traffic-dependent parameters are from [15]. The channelmodel parameters are taken from [30] and [5]. Results areobtained for a signal-to-noise ratio of γ = 0 dB.Fig. 3a shows the EE as a function of M and N when K =10 . We see that there is a global maximizer for ( M ⋆ , N ⋆ ) =(19 , to which corresponds an EE ⋆ = 620 Mbit/Joule anda throughput of . Gbit/s per small-cell BS. The total powerconsumed by circuitry is approximately P CP = 290 W. Thesequential optimization algorithm described in Section IV con-verges after a few iterations to the global optimizer validating(16). As seen, the optimal configuration is characterized bya relatively small N ⋆ = 30 , which is slightly larger than thenumber of served small cells, i.e. K = 10 . In other words, theoutput of the optimization problem suggests to use a numberof BS antennas that is on the same order of magnitude of K .This is in contrast to what it is usually required in mmWavecommunications for maximal spectral efficiency, namely, alarge antenna array at both sides of the link to cope withthe severe propagation conditions. To be energy-efficient, theso-called doubly massive MIMO paradigm requires eitherbetter beamforming schemes (increasing the throughput) ormore power efficient electronic devices (reducing the powerconsumption). This latter case is investigated in Fig. 3b inwhich the power consumed by front-end devices is decreasedby an order of magnitude, both at the BS ( p BSFE ) and at thesmall-cell BSs ( p SCFE ). We see that in this case a doubly massive CMOS technology promises higher levels of integration and reduced costwith respect to other solutions on the market such as GaAs and InP [29]. In literature doubly massive MIMO is referred to a system equipped withvery large antenna arrays at both transmitter and receiver.
Number of antennas N Number of antennas M EE [ M b it/ J ou l e ] EE-optimal EE ⋆ =620 Mbit/Joule , ( M ⋆ ,N ⋆ )=(19 , Sequential optimization (a) p BSFE and p SCFE as in Table I
Number of antennas N Number of antennas M EE [ M b it/ J ou l e ] EE-optimal EE ⋆ =710 Mbit/Joule , ( M ⋆ ,N ⋆ )=(108 , Sequential optimization (b) p BSFE and p SCFE as in Table I scaled by a factor × Fig. 3: Energy Efficiency [Mbit/Joule] for different combina-tion of M and N (with K = 10 ).MIMO setup with ( M ⋆ , N ⋆ ) = (108 , naturally arises atthe EE-optimal. The throughput is also increased by a factor . × with respect to the EE-optimal in Fig. 3a. Based on theabove results, it follows that, to improve the EE and throughputof mmWave communications, the hardware components (suchas PSs, LNAs and HPAs) have to be more efficient than todays.VI. E XTENSION TO NL O S CHANNELS
In this section, we investigate to what extent the majorconclusions can be extended to a NLoS scenario. . Network model
We adopt a time-invariant clustered channel model com-posed of a LoS path and N cl scattering clusters, each onecontributing with N r rays accounting for the NLoS component.This leads to the following channel matrix H k ∈ C N × M between the BS and small-cell BS k : H k = 1 √ N cl N r N cl X i =1 N r X j =1 √ α i,j,k a N ( φ i,j,k ) a H M ( θ i,j,k ) + I LoS ( d k ) √ α k a N ( φ k, LoS ) a H M ( θ k, LoS ) (31)where φ i,k and θ i,k are the mean AoD and AoA of eachlink between BS and the i -th scatterer. The angle spreadwithin each cluster is also taken into account by usingLaplacian distribution, φ i,j,k ∼ L ( φ i,k , µ i,k ) and θ i,j,k ∼L ( θ i,k , µ i,k ) . The parameter α i,j,k includes both the small-scale and the large-scale fading effect and is computed as α i,j,k = ˜ α i,j,k − l i,k, dB / with l i,k, dB as in (2) and ˜ α i,j,k accounting or the small-scale effects. The set of NLoS dis-tances can be evaluated by geometrical considerations as d i,k = d cl i,k + q ( d cl i,k sin ¯ φ i,k ) + ( d k − d cl i,k cos ¯ φ i,k ) (32)where d cl i,k and d k are the distances BS-cluster i (whenpointing small cell k ) and BS-small cell k , respectively and ¯ φ i,k = φ i,k − φ k, LoS , ¯ θ i,k = θ i,k − θ k, LoS . Besides, in theLoS component, I LoS ∼ B ( p ( d k )) is a Bernoulli randomvariable indicating the presence or not of the LoS link .Unlike the NLoS component, θ k, LoS and φ k, LoS are relatedas θ k, LoS = mod( π + φ k, LoS , π ) . We refer to [30] and [3] forfurther details. Hereafter, to dimension the precoder/combinerwe use the same eigenmode beamforming approach used inSection II, in the analog domain, along with a digital ZFprecoder. In particular, let H H k = U k Σ k V H k be the singularvalue decomposition (SVD) of H H k , the k-th user precodingand combining vectors, f RF ,k and w k , are chosen as thecolumns of the matrices V k and U k corresponding to thelargest eigenvalue of Σ k , i.e. v k, and u k, . We then projectthe beamforming matrices F RF and W onto the analog set S p,q = { X ∈ C p × q : | X i,j | = 1 , ( i, j ) = { p } × { q }} . Thissimply results in scaling each entry of those matrices by itsmagnitude [8]. The precoder F BB is designed according to aZF criterion to cope with the effective interference after analogprecoding-combining. B. Numerical results
Fig. 4a shows numerically how the EE behaves as a functionof M and N using the NLoS channel model described above.The optimal operating point is found at ( M ⋆ , N ⋆ ) = (5 , to which corresponds an EE = 709 Mbit/Joule, an aggregatethroughput and circuit power consumption are respectively . Gbit/s per small-cell BS and
W. The above networkconfiguration is far from being considered as doubly-massiveMIMO. This supports our conclusion that such systems, when Reasonably, p ( d k ) i.e. the probability to have LoS link, it is modeled witha monotonic non-increasing function of its argument. Number of antennas N Number of antennas M EE [ M b it/ J ou l e ] EE-optimal EE ⋆ =709 Mbit/Joule , ( M ⋆ ,N ⋆ )=(5 , (a) Hybrid precoder Number of antennas N Number of antennas M EE [ M b it/ J ou l e ] EE-optimal EE ⋆ =714 Mbit/Joule , ( M ⋆ ,N ⋆ )=(1 , (b) Fully-digital precoder Fig. 4: Energy Efficiency [Mbit/Joule] for different combina-tion of M and N (with K = 10 ) for hybrid and fully-digitalprecoder.used with hybrid architectures, are not optimal from an EEperspective. Fig. 4b illustrates the EE of a fully-digital system,which applies the ZF precoder entirely in the baseband, thatis F RF F BB = F = ¯ H † . In addition, to fairly compare theperformance of the fully-digital to that of the hybrid schemein Fig. 1, constant transmit power at BS is ensured, i.e. k x k = k s k . The transmitted vector of symbols must bechanged accordingly so as s ′ = ( M ⊗ I K ) s . At the smallcell side, linear combining is performed by matching themost significant left eigenvector of the channel w k = u k, associated to the highest eigenvalue. Fig. 4b further validatesthe tendency encountered for the hybrid system, which is toavoid the use of large arrays at both network sides. Here,ABLE II: Power consumption of the different components atthe operating point ( M ⋆ , N ⋆ ) with P FIX = 50 W . Power parameters Hybrid Fully-digital P FIX
68% 17% P RF
5% 16% P FE
24% 65% P LP
3% 2% the EE-optimal point is at ( M ⋆ , N ⋆ ) = (1 , achieving athroughput of . Gbit/s per small-cell BS with
W ofconsumed power. Although the precoders perform similarly,the hybrid solution leads to a smoother EE function that ispreferable for its robustness to system changes. Moreover,Table II shows how much the circuit power terms contributeto the overall consumed power at the EE-optimal, both for thehybrid and fully-digital case. As we can see, in the hybrid case,the major contribution comes from the fixed power, while inthe fully-digital one it comes from the power drawn by the FEchain at the BS. This is due to the high power required by oneDAC per antenna. Those costs scale linearly with N insteadof with K , becoming prohibitive in the large array domain.VII. C ONCLUSIONS
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